| Chapter 1 Fundamental Number-Theoretic Algorithms 1.1 Introduction 1.1.1 Algorithms 1.1.2 Multi-precision 1.1.3 Base Fields and Rings 1.1.4 Notations 1.2 The Powering Algorithms 1.3 Euclid's Algorithms 1.3.1 Euclid's and Lehmer's Algorithms 1.3.2 Euclid's Extended Algorithms 1.3.3 The Chinese Remainder Theorem 1.3.4 Continued Fraction Expansions of Real Numbers 1.4 The Legendre Symbol 1.4.1 The Groups (Z/nZ)* 1.4.2 The Legendre-Jacobi-Kronecker Symbol 1.5 Computing Square Roots Modulo p 1.5.1 The Algorithm of Tonelli and Shanks 1.5.2 The Algorithm of Cornacchia 1.6 Solving Polynomial Equations Modulo p 1.7 Power Detection …… 1.8 Exercises for Chapter 1 Chapter 2 Algorithms for Linear AQlgebra and Lattices 2.1 Introducion 2.2 Linear Algebra Algorithms on Square Matrices 2.3 Linear Algebra on General Matrices 2.4 Z-Modules and the Hermite and Smith Normal Forms 2.5 Generalities on Lattices 2.6 Lattice Reducion Algorithms 2.7 Applications of the LLL Algorithm 2.8 Exercises for Chapter 2 Chapter 3 Algorithms on Polynomials 3.1 Basic Algorithms 3.2 Euclid's Algorithms for Polynomials 3.3 The Sub-Resultant Algorithm 3.4 Factorization of Polynomials Modulo p 3.5 Factoriztion of Polynomials over Z or Q 3.6 Additiional Polynomial Algoritms 3.7 Exercises for Chapter 3 Cahpter 4 Algorithms for Algebraic Number Theory I Cahpter 5 Algorithms for Quadratic Fields Cahpter 6 Algorithms for Algebraic Number Theory II Cahpter 7 Introducion to Elliptic Curves Cahpter 8 Factoring in the Dark Ages Cahpter 9 Modern Primality Tests Cahpter 10 Modern Factoring Methods Appendix A Packages for Number Theory Appendix B Some Useful Tables Bibliography Index |
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